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Relativistic Hydrodynamics$

Luciano Rezzolla and Olindo Zanotti

Print publication date: 2013

Print ISBN-13: 9780198528906

Published to Oxford Scholarship Online: January 2014

DOI: 10.1093/acprof:oso/9780198528906.001.0001

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(p.665) Appendix C Notable Tensors

(p.665) Appendix C Notable Tensors

Source:
Relativistic Hydrodynamics
Publisher:
Oxford University Press

We report below the most relevant tensors, in their component notation, that have been used in the derivation of the relativistic-hydrodynamic equations.

C.1 Relativistic expressions

A detailed discussion of the properties of the tensors presented here can be found in Sections 3.1.1 and 3.7.4.

Θ := μ u μ , ( expansion scalar )
(C.1)
a μ := u ν ν u μ , ( kinematic acceleration four-vector )
(C.2)
ω μ ν := [ ν u μ ] + a [ μ u ν ] , ( kinematic vorticity tensor )
(C.3)
ω μ := 1 2 η μ ν α β ω α β u ν , ( kinematic vorticity four-vector )
(C.4)
Ω μ ν := 2 [ ν h u μ ] , ( vorticity tensor )
(C.5)
Ω μ μ = 0 , ( trace of vorticity tensor )
(C.6)
Ω μ := 1 2 η μ ν α β Ω α β u ν , ( vorticity four-vector )
(C.7)
ω := ( 1 2 ω μ ν ω μ ν ) 1 2 , ( vorticity scalar )
(C.8)
σ μ ν := ( μ u ν ) + a ( μ u ν ) 1 3 Θ h μ ν , ( shear tensor )
(C.9)
σ μ μ = 0 , ( trace of shear tensor )
(C.10)
σ := ( 1 2 σ μ ν σ μ ν ) 1 2 , ( shear scalar )
(C.11)
ν u μ = ω μ ν + σ μ ν + 1 3 Θ h μ ν a μ u ν , ( irreducible decomposition )
(C.12)
h μ ν := g μ ν + u μ u ν , ( projection tensor )
(C.13)
T μ ν = e u μ u ν + ( p + Π ) h μ ν + q μ u ν + q ν u μ + π μ ν ,
(C.14)
= ( e + p ) u μ u ν + p g μ ν if q μ = 0 = Π = π μ ν , ( energy–momentum tensor )
(C.15)
q μ := h α μ T α ν u ν , ( energy flux )
(C.16)
p + Π := 1 3 h μ ν T μ ν , ( isotropic pressure and viscous bulk pressure )
(C.17)
π μ ν := h α μ h β ν T α β ( p + Π ) h μ ν , ( anisotropic shear tensor )
(C.18)
Π = ζ Θ ,
(C.19)
q μ = κ T ( h μ ν ν ln T + a μ ) ,
(C.20)
π μ ν = 2 η σ μ ν ,
(C.21)

(p.666) where η and ζ are the shear viscosity and the bulk viscosity, respectively.

C.2 (p.667) Newtonian expressions

To aid comparison with classical textbooks, e.g., Landau and Lifshitz (1980), we report below the Newtonian expressions of the most relevant hydrodynamic tensors. A detailed discussion of the properties of the tensors presented here can be found in Section 2.2.6.

θ := k v k , ( expansion scalar )
(C.22)
ω N i := ϵ i j k j v k , ( vorticity vector )
(C.23)
Λ i j := 1 2 ( i v j + j v i ) , ( strain tensor )
(C.24)
Λ i i = θ , ( trace of strain tensor = expansion )
(C.25)
Λ i j TF := Λ i j 1 3 θ δ i j , ( shear tensor )
(C.26)
P i j := p δ i j 2 η Λ i j TF , ( pressure tensor )
(C.27)
P i i = 3 p , ( trace of pressure tensor )
(C.28)
S i j := 2 η Λ i j TF + ζ θ δ i j = P i j + ( p + ζ θ ) δ i j , ( viscous stress tensor )
(C.29)
S i i = 3 ζ θ , ( trace of viscous tensor )
(C.30)
Σ i j := p δ i j + S i j = P i j + ζ θ δ i j , ( stress tensor )
(C.31)
Σ i i = 3 ( ζ θ p ) . ( trace of stress tensor )
(C.32)